# Full text of "The Wealth and Well-Being Curves, and Political Freedom: A Mathematical Analysis: Part 1"

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```The Wealth and Weil-Being Curves, and Political
Freedom - A Mathematical Analysis: Part 1

By Patrick Bruskiewich
Mathematisches Institut der Vancouver, BC

Abstract

In a recent newspaper the author came across a scatter plot of Wealth (GDP) versus Well-
Being. The distribution is similar to those found in Learning Theory and in a number of
Applied Mathematics models. The scatter plot begs a mathematical analysis as well as a
correlation to socio-economic measures.

1.0 The Wealth and the Weil-Being Curve

In a recent edition of a Canadian newspaper The National Post the author came across an
article about the puzzling angst of the People of France. With this March 26, 2013 article
was included a distribution that plotted Gross Domestic Product as an independent
variable with an implied correlation to a dependent "subjective Well-Being Index" (refer
to Fig. 1 : GDP versus Well-Being Index).

The distribution shows a number of indicators that relate or correlate to the "subjective
Well-Being Index", and representative economic systems. The scatter plot came
complete with a curve (denoted red in the article) with no description of the functional
form the fit. The distribution was sourced from the Paris School of Economics.

A plethora of curves can be fitting to this distribution depending on the variables used.
Perhaps, to a first approximation a restricted growth algorithm, or Mitscherlich
relationship, could be used to describe the distribution. [1]

■ LATIN AMERICA o FORMER COMMUNIST

4.5

-LB
-1.5
-Z.B
-2.5

Puerto Ricn

o Ireland

New

Zealand

Denmark
Iceland .>-.
A Switzerland

Netherlands •

Sweden -a * united States^

Australia -o J^ mximbowg^
Britain BeJoium
Finland -o 6 A, o- Austria N°™ay-o

h

India
c

„ !■ /y o Serbia
BurkinaXy'

cardan

Tanzania
^Azerbaijan ^^

Rwanda o- Estonia

Latvia-Q o- Lithuania

Georgia - , jv Russia

j. Bulgaria -do

Alhania-o Romania

J Belarus^ ^ Jkraine

O- Armenia

9
ZSmbatae

Curve with article
Fitted SWI Curve

~i ; 1 i

5 10 15 20

GDP PER CAPITA, IN THOUSANDS OF DOLLARS

25

3D

SOUH.CK: I'AKIS SCHOOL ("Jl-' ECONOMICS

JONATI [ON IlIVAiT / NATIONAL POST

Fig. 1: GDP versus Weil-Being Index

2.0 Restrictive Growth and the Mitscherlich Relationship

The Mitscherlich Relationship describes a system governed by the differential equation

dw

dt

= k(B-w)

where k is a positive constant which determines how fast the differential tends to zero,
and where B represents an upper bound. There exists an upper bound B for the
productivity of the system (the line y = B is the asymptote).

No crop, industry or population can grow or flourish indefinitely. There are limitations
set by shortages of raw materials, productivity of land and by some other sort of control
mechanism, such as limits to demand, inherent economic efficiencies and the manner in
which decisions are made or populations are governed.

For example, there is a relationship between fertilizer use and crop yield. To first
approximation, the function y can be considered as a monotonically increasing function.
If x where the amount of fertilizer applied to the cultivation of a crop, the yield y cannot
be raised indefinitely by the application of more and more fertilizer. All things being
equal, the land would, with time, become limited in its crop productivity as the soil
becomes saturated with fertilizer.

For such a system a workable approximation to the output or productivity of the land is
given by the Mitscherlich Relationship,

y = B(l-e- b )

In analyzing the locus of points that is graphed with the original article, the author was
able to fit a simple Mitscherlich Relationship to the red curve which has the form

SWI = 7r(\-e kw )

l

where k = — , which appears to be a defining feature on the left hand side of the
distribution and appears to be an inflection point for the distribution.

The plot of the Subjective Well-Being Index (SWI) function is the blue line in Fig. 1.
The graph and the SWI fit (blue curve) begs an in-depth mathematical analysis, as well as
a correlation on socio-economic indicators.

3.0 Political Freedom as a Significant Socio-Economic Measure

Based on a careful reading of history, it is evident that there is a continuum of Political
Freedom that spans from Totalitarianism at one extreme, to a full and open Liberal
Democratic Institutions and the Ecole Suisse at the other extreme. An index that runs
from to 3 can be applied to the Continuum of Political Freedom (refer to Table 1 :
Continuum of Political Freedom)

0.5

1.0

1.5

2.0

2.5

3.0

Totalitarian

Neo-
Communism

Autocratic

Oligarchy

Capitalism

Liberal
Democracy

Ecole
Suisse

Table 1: Continuum of Political Freedom

To more fully represent a correlation between Wealth and Well-Being, while applying a
measure of the Continuum of Political Freedom (0 < m < 3), we must look beyond the
simplicity of the Mitscherlich Relationship.

4.0 Modeling SWI using the Logistic Equation

It would seem reasonable to develop a model that is independent of monetary measures
that depend currency exchange are are subject to inflation and other competitive factors.

Beyond the Mitscherlich Relationship, restrictive growth can be described by a
differential equation of the expanded form

where X is a positive coupling constant . The solution for this expanded expression is

B

y ~\ + ke- m

where k has a positive value. For the case k = 1 and e ~ lBt « 1 we get y « B 1 1 - e~ xm j .
Based on the SWI functional fit (above)

10 IOtt

Using a scaling factor, let

t = l0m

Then the coupling factor becomes

7t

5.0 Subjective Well-Being and Political Freedom

Consider a logistic function of the more complete form

dy
dm

= A'(A-y)(B-y)

where m is the Continuum of Political Freedom ( < m < 3 ) outlined above. The
solution for this more general expression is the logistic equation

y = A +

(B-A)

l + ke

A'(B-A)m

It is worth noting that n appears as the factor B in the Mitscherlich Relationship. Our
more developed model should involve universal measures such a n . We can do this
by choosing a function of the form for the Subjective Well-Being Index,

SWI = -2.0 +

2n

\ + ke

-2.0m

Such a function should describe a span from a well-being index of -1.5 to 4.25 from
whence we conjecture that the factor k is of the form

k = 7u { *- x) * 11.606

The inflection point is at

d

r ^,\

dm

dy ]_ d ro ,
V dm ) dm

\A'(A-y)(B-y)] =

which occurs at the value

y = \(B + A)

which yields

y = (7U-2) = \. 14159

For SWI as a function of Political Freedom ( < m < 3 ) we find the following (refer to
Table 2: SWI as a function of the Political Freedom m):

Continuum Measure
m

Political Freedom

Subjective Well-
Being Index

Efficiency

Totalitarian

-1.5

-0.37

0.5

Neo- Communism

-0.81

-0.20

1.0

Autocratic

0.44

0.11

1.5

Oligarchy

1.98

0.48

2.0

Capitalism

3.18

0.78

2.5

Liberal Democracy

3.82

0.93

3.0

Ecole Suisse

4.10

1.0

Table 2: SWI as a function of the Political Freedom Measure m

A graph of the Subjective Well-Being Index (SWI) as a function of the Political Freedom
Measure m is given in Fig. 2: The SWI as a function of the Political Freedom Measure m
with some representative countries plotted on the distribution.

It is clearly evident the more open forms of government not only provide for a better
sense of Well-Being and Quality of Life they also, by the nature of the Logistic equation,
provide for greater opportunity for growth and for economic efficiency.

SWI as a function of m

-'

Denmark

Suisse

Ideal

Brazil

^*

-*S nl j, * USA

Taiwan

_*_ •» "OTC

Singapore wf

Tig Spain W Japan

China* Poland

^J" South Korea

South Africa
♦ India W Hungary

5/

Russia

' :

■JJ 1 Ukraine
•ff North Korea

m (Political Frasflom)

35 1

Fig. 2: The SWI as a function of the Political Freedom Measure m

A measure of efficiency is the ratio of the SWI for a Political Freedom Measure m
compared to the Ideal Measure at m = 4.0 (refer to Fig. 3: Economic Efficiency as a
Function of the Political Freedom Measure m).

Clearly the Totalitarian and the neo-Communism Regime are less than efficient in their
provisioning of goods and services compared to the Open Capitalism, the countries with
Parliamentary and Liberal Democratic ideals, and the Ecole Suisse.

6.0 Learning, Training and Development

In a real sense the SWI function describes the base and the enlightened extremes of
human behaviour. One could speculate that the Continuum Measure m is a measure of
Informed -Consent, and a measure s which represents an Efficiency of Governance.

Bf iciency as a f Lncii on of m

120
1.00
30
60

fc 040

020

^ 000

-3 20

-: io

-0 60

m {Ported fT««dom|i

\$ulm

IJmI I

CW, ^*^^"*1HA

s

/ ^L South *V#»

>

1 1 aiy^

1 1,5 2 2 5 J

5 5

:

*m»»

WeithKfrru

Fig. 3: Economic Efficiency as a Function of the Political Freedom Measure m

For instance, for a measure of 1 < / < 3 and 1 < C < 3 , a simple model for m would be

9 V ;

and where < s < 1 . The form of the SWI Function is similar to mathematical functions
that describe Learning, Training and Development. This may be touched upon in a
subsequent paper.

You will note the anomalous nature of Communist North Korea under Kim and Neo-
Communist Russia under Putin, compared to for instance the People's Republic of China.

References:

[1] E. Batschelet, Introduction to Mathematics for Life Scientists, Springer- Verlag,

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